Variational approach for edge-preserving regularization using coupled PDEs

This paper deals with edge-preserving regularization for inverse problems in image processing. We first present a synthesis of the main results we have obtained in edge-preserving regularization by using a variational approach. We recall the model involving regularizing functions phi and we analyze the geometry-driven diffusion process of this model in the three-dimensional (3-D) case. Then a half-quadratic theorem is used to give a very simple reconstruction algorithm. After a critical analysis of this model, we propose another functional to minimize for edge-preserving reconstruction purposes. It results in solving two coupled partial differential equations (PDEs): one processes the intensity, the other the edges. We study the relationship with similar PDE systems in particular with the functional proposed by Ambrosio-Tortorelli in order to approach the Mumford-Shah functional developed in the segmentation application. Experimental results on synthetic and real images are presented.

[1]  Paolo Nesi,et al.  Variational approach to optical flow estimation managing discontinuities , 1993, Image Vis. Comput..

[2]  Michel Barlaud,et al.  Deterministic edge-preserving regularization in computed imaging , 1997, IEEE Trans. Image Process..

[3]  D. Mumford,et al.  Optimal approximations by piecewise smooth functions and associated variational problems , 1989 .

[4]  Jitendra Malik,et al.  Scale-Space and Edge Detection Using Anisotropic Diffusion , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[5]  Michel Barlaud,et al.  Two deterministic half-quadratic regularization algorithms for computed imaging , 1994, Proceedings of 1st International Conference on Image Processing.

[6]  Michel Barlaud,et al.  Segmentation and edge-preserving restoration , 1997, Proceedings of International Conference on Image Processing.

[7]  Mark Nitzberg,et al.  Nonlinear Image Filtering with Edge and Corner Enhancement , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[8]  P. Lions,et al.  Image recovery via total variation minimization and related problems , 1997 .

[9]  M. Barlaud,et al.  Nonlinear image processing: modeling and fast algorithm for regularization with edge detection , 1995, Proceedings., International Conference on Image Processing.

[10]  Michel Barlaud,et al.  Nonlinear regularization using constrained edges in image reconstruction , 1996, Proceedings of 3rd IEEE International Conference on Image Processing.

[11]  P. Green.  Bayesian reconstructions from emission tomography data using a modified EM algorithm. , 1990, IEEE transactions on medical imaging.

[12]  Bart M. ter Haar Romeny,et al.  Geometry-Driven Diffusion in Computer Vision , 1994, Computational Imaging and Vision.

[13]  L. Vese,et al.  A Variational Method in Image Recovery , 1997 .

[14]  P. Lions,et al.  Image selective smoothing and edge detection by nonlinear diffusion. II , 1992 .

[15]  L. Ambrosio,et al.  Approximation of functional depending on jumps by elliptic functional via t-convergence , 1990 .

[16]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[17]  Andrew Blake,et al.  Visual Reconstruction , 1987, Deep Learning for EEG-Based Brain–Computer Interfaces.

[18]  Nicholas Ayache,et al.  Directional Anisotropic Diffusion Applied to Segmentation of Vessels in 3D Images , 1997, Scale-Space.

[19]  C. Vogel,et al.  Analysis of bounded variation penalty methods for ill-posed problems , 1994 .

[20]  Ken D. Sauer,et al.  A generalized Gaussian image model for edge-preserving MAP estimation , 1993, IEEE Trans. Image Process..

[21]  Donald Geman,et al.  Constrained Restoration and the Recovery of Discontinuities , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[22]  Niklas Nordström,et al.  Biased anisotropic diffusion: a unified regularization and diffusion approach to edge detection , 1990, Image Vis. Comput..

[23]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[24]  Guillermo Sapiro,et al.  From active contours to anisotropic diffusion: connections between basic PDE's in image processing , 1996, Proceedings of 3rd IEEE International Conference on Image Processing.

[25]  Luigi Ambrosio,et al.  ON THE APPROXIMATION OF FREE DISCONTINUITY PROBLEMS , 1992 .

[26]  Wenyuan Xu,et al.  Behavioral analysis of anisotropic diffusion in image processing , 1996, IEEE Trans. Image Process..

[27]  H. Fédérer.  Geometric Measure Theory , 1969 .

[28]  T. Hebert,et al.  A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors. , 1989, IEEE transactions on medical imaging.

[29]  Jayant Shah,et al.  A common framework for curve evolution, segmentation and anisotropic diffusion , 1996, Proceedings CVPR IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[30]  D. Dobson,et al.  Convergence of an Iterative Method for Total Variation Denoising , 1997 .

[31]  Riccardo March,et al.  Visual reconstruction with discontinuities using variational methods , 1992, Image Vis. Comput..